googologywikiaorg-20200223-history
User blog:Undeadlift/Supernova Array
The Supernova Array is a method of creating very large numbers using recursion of Up-Arrow Notation in a simple linear notation. It is loosely based on the Ultra-Factorial Funcional Array. Definitions and Examples *Base Function: \(S(a,b) = a\uparrow^{(b)}a\), where \(b\) is the number of \(\uparrow\)'s. *\(S(a,b,0) = S(a,b)\) *\(S(a,b,c) = S(a,S(a,b),c-1)\) *\(S(a,b,c,0) = S(a,b,c)\) *\(S(a,b,c,d) = S(a,b,S(a,b,c),d-1)\) *\(S(a,b,c,...x,y,z) = S(a,b,c,...x,S(a,b,c,...x,y),z-1)\) In general, replace the second to the last input (\(y\)) with the entire function excluding the last input (\(z\)), then subtract 1 from the last input. Do the same for the function within, and the subsequent function, and the function after that, etc. until you reach the base function \(S(a,b)\). \(S(1,1) = 1\uparrow 1 = 1\) \(S(2,2) = 2\uparrow \uparrow 2 = 4\) \(S(3,3) = 3\uparrow \uparrow \uparrow 3\) = tritri \(S(10,10) = 10\uparrow^{(10)} 10\) = tridecal \(S(3,1,1) = S(3,S(3,1),0) = S(3,S(3,1)) = S(3,3\uparrow 3) = S(3,27)\) \(S(3,3,1) = S(3,S(3,3)) = S(3,tritri)\) \(S(10,10,1) = S(10,S(10,10)) = S(10,tridecal)\) \(S(3,3,3) = S(3,S(3,3),2) = S(3,tritri,2) = S(3,S(3,tritri),1) = S(3,S(3,S(3,tritri)),0)\) \(S(3,4,63) = S(3,S(3,4),62) = S(3,S(3,S(3,4)),61)...\) = Graham's Number Extended Notations In extended notations, parentheses are added into the array in order to apply systems of recursion. *\(S((a,b)) = S(a,a,a,...a)\), where there are \(b\) number of \(a\)'s *\(S((a,b,c,...x,y,z)) = S((a,b,c,...x,S(a,b,c,...x,y),z-1))\) *\(S(((a,b))) = S((a,a,a...,a))\), where there are \(b\) number of \(a\)'s *\(S(((a,b,c,...x,y,z))) = S(((a,b,c,...x,S(a,b,c,...x,y),z-1)))\) In general, \(S(a,b)_n = S(a,a,a,...a)_{n-1}\), where there are \(b\) number of \(a\)'s and \(n\) number of paretheses. Alternatively, \(S(((a,b)))\) can be written as \(S(a,b)_3\). There may be cases wherein a simplified notation may be desirable, as in \(S(a,b)_{S(a,b)}\). In this case, a subscript is added to S, such that the aforementioned function can be rewritten as \(S_1(a,b)\). In general, *\(S_0(a,b) = S(a,b)\) *\(S_{(n+1)}(a,b) = S(a,b)_{(S_n(a,b))}\), where there are \(S_n(a,b)\) number of parentheses in \(S(((...(a,b)))...)\) *\(S_n(a,b,c,...x,y,z) = S_n(a,b,c,...x,S(a,b,c,...x,y),z-1)\) Likewise, a simplified notation may be desired for functions such as \(S_{S(a,b)}(a,b)\). A superscript is instead assigned to S, such that the aforementioned function may be rewritten as \(S^1(a,b)\). In general, *\(S^0(a,b) = S(a,b)\) *\(S^{(n+1)}(a,b) = S_{(S^n(a,b))}(a,b)\) *\(S^n(a,b,c,...x,y,z) = S^n(a,b,c,...x,S(a,b,c,...x,y),z-1)\) Furthermore, \(S^{S(a,b)}(a,b)\) could also use a more simplified notation. A subscript is then added BEFORE S, such that the aforementioned function may be rewritten as \(_1S(a,b)\). In general, *\(_0S(a,b) = S(a,b)\) *\(_{(n+1)}S(a,b) = S^{(_nS(a,b))}(a,b)\) *\(_nS(a,b,c,...x,y,z) = _nS(a,b,c,...x,S(a,b,c,...x,y),z-1)\) Finally, a function such as \(_{(S(a,b))}S(a,b)\) would also benefit from simplification. A superscript is then added BEFORE S, such that the aforementioned function may be rewritten as \(^1S(a,b)\). In general, *\(^0S(a,b) = S(a,b)\) *\(^{(n+1)}S(a,b) = _{(^nS(a,b))}S(a,b)\) *\(^nS(a,b,c,...x,y,z) = ^nS(a,b,c,...x,S(a,b,c,...x,y),z-1)\) However, the function's limit has been reached with the pre-superscript. In order to further expand the function, a new more succint formulation has to be added - \(S(a,b)n,k\). *\(S(a,b)_n\) can be rewritten as \(S(a,b)n,0\), *\(S_n(a,b)\) can be rewritten as \(S(a,b)n,1\), *\(S^n(a,b)\) can be rewritten as \(S(a,b)n,2\), and so on. In general, *If \(k = 0\), \(S(a,b)_n\ = S(a,b)n,0 = S(a,a,a,...a)n-1,0\), where there are \(b\) number of \(a\)'s *If \(k > 0\) and \(n = 0\), \(S(a,b)0,k = S(a,b)\) *If \(k > 0\) and \(n > 0\), \(S(a,b)n,k = S(a,b)[S(a,b)n-1,k,k-1]\) Although this function can already produce extremely huge numbers, it can still be further extended as follows: \(S(a,b)n,k,0 = S(a,b)n,k\) \(S(a,b)n,k,p = S(a,b)[n,S(a,b)n,k,p-1]\) \(S(a,b)n,k,p,...x,y,z = S(a,b)[n,k,p,...x,S(a,b)n,k,p,...x,y,z-1]\) \(S(a,b)[ n,k ] = S(a,b)n,n,n,...n\), where there are \(k\) number of \(n\)'s \(S(a,b)n,k1,0 = S(a,b)n,k0,s = S(a,b)n,k\) \(S(a,b)n,kr,0 = S(a,b)n,n,n,...nr-1,0\), where there are \(k\) number of \(n\)'s \(S(a,b)n,kr,s = S(a,b)n,k[S(a,b)n,kr-1,s,s-1]\) \(S(a,b)n,kr,s,0 = S(a,b)n,kr,s\) \(S(a,b)n,kr,s,t = S(a,b)n,k[r,S(a,b)n,kr,s,t-1]\) \(S(a,b)n,kr,s,t,...x,y,z = S(a,b)n,k[r,s,t,...x,S(a,b)n,kr,s,t,...x,y,z-1]\) \(S0(a,b) = S(a,b)\) \(S\gamma(a,b) = S-1(a,b)a,aa,aa,a...a,a\), where there are \(b\) number of \(a\)'s \(S\gamma(a,b)n,k = S\gamma(a,b)[S(a,b)n-1,k,k-1]\) Etc. Category:Blog posts